Lesson 8.1

Ratio: a ratio is a quotient of two numbers.

a:b a to b a÷b

a

b

Always given in lowest terms.

Slope of a line is a ratio between two points. (rise over run)

Proportions: two or more ratios set equal to each other.

a

b

c

d=

a:b = c:d

a is the first term

b is the second term

c is the third term

d is the fourth term

Product and Ratio Theorems

In a product containing four terms:

First and fourth terms are the extremes.

Second and third terms are the means.

Theorem 59: In a proportion, the product of the means is equal to the product of the extremes. (means-extremes product theorem.)

a

b

c

d= ad = bc

If they aren’t equal, then the ratios aren’t in proportion.

Theorem 60: If the product of a pair of non-zero numbers is equal to the product of another pair of non-zero numbers, then either pair of numbers may be made the extremes, and the other pair the means, of a proportion. (means-extremes ratio theorem.)

This theorem is harder to state than to use!Given: pq = rsThen:

p

r

s

q

p

s

r

q

r

p

q

s= = =

pq = rs pq = rs pq = rs

These proportions are all equivalent since their cross products are equivalent equations.

In a mean proportion, the means are the same.

4

16

1

4=

a

x

x

r=

4 is the geometric mean

x is the geometric mean

Definition: If the means in a proportion are equal, either mean is called a geometric mean or mean proportional between the extremes.

Find the arithmetic & geometry means between 3 and 27.

Arithmetic mean:

2

273

= 15

Geometric mean:

3

x

x

27=

x2 = 81

x = 9

Solve:

7

14

3

x=

You might want to reduce the fraction first.

7x = 42

x = 6

2

3

4

x=

2x = 12

x = 6

Find the fourth term (sometimes called the fourth proportional) of a proportion if the first three terms are 2, 3, and 4.

Find the mean proportional(s) between 4 and 16.

4

x

x

16=

x2 = 64

x = 8

If we are looking for the length of a segment, then only the positive number works.

If 3x = 4y, find the ratio of x to y.

Make x and 3 the extremes and y and 4 the means.

3x = 4y

x

y

4

3=

Is

a

b

x

y= equal to

a 2bb

x 2yy

= ?

Cross multiply and simplify both sets.

ay = bx

b(x-2y) = y(a-2b)

bx-2by = ay-2by

bx = ay

Yes, they are equal.